= Two-sample FPCA mean statistic
{c}
Let $X_1,\ldots,X_n$ and $Y_1,\ldots,Y_n$ be two <independent> samples of <independent and identically distributed> <Hilbert-space-valued random variables>, with finite second <moments> and <covariance operators> $C_X,C_Y$. To test equality of their <expectations>, use the pooled <covariance operator> $C=(C_X+C_Y)/2$. This choice works even when the two population <covariance operators> differ: under the <null hypothesis>, the <covariance operator> of $\sqrt{n/2}(\overline X-\overline Y)$ is exactly $C$.
Let $(\widehat\lambda_j,\widehat\phi_j)$ estimate the <eigenvalues> and orthonormal <eigenvectors> of $C$, for example from the average of the two within-sample empirical <covariance operators>. For fixed $K$, positive retained <eigenvalues>, a separated retained <eigenspace> and suitable <statistical consistency> of these estimates,
$$
T_{n,K}=\frac n2\sum_{j=1}^K
\frac{\langle\overline X-\overline Y,\widehat\phi_j\rangle^2}{\widehat\lambda_j}
\xrightarrow{d}\chi_K^2.
$$
The proof applies the <Hilbert-space central limit theorem> to the difference of <sample means>, then standardizes its first $K$ coordinates and uses <Slutsky's theorem>. A sufficient simple-eigenvalue condition is $\lambda_1>\cdots>\lambda_K>\lambda_{K+1}$ with $\lambda_K>0$.<Estimation of the Mean of Functional Time Series and a Two-Sample Problem by Lajos Horváth and others>[Section 3, equation (3.6) and Theorem 5]{ref}
For unequal sample sizes $n,m$, the scaling becomes $nm/(n+m)$ and the relevant population <covariance operator> is $\{mC_X+nC_Y\}/(n+m)$. Thus a sample-size-weighted pooled estimate cannot automatically be reused for unequal sizes and unequal <covariance operators>: its weights must match the <variance> of the difference being tested.
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